1. Field of the Invention
The subject invention is generally related to wind power turbines and is specifically directed to a family of airfoil configurations for optimizing the performance of the wind turbine.
2. Discussion of the Prior Art
Wind power turbines are well known. The turbine blades or airfoils are one of the primary factors in determining the efficiency of the system and thus are a critical factor in optimizing performance. Typically, the turbine rotor blade design proceeds by first identifying a family of airfoils to be employed and then determining the optimum spanwise distribution of solidity and twist in order to optimize the power coefficient at each spanwise location. This procedure often does not result in the structurally optimal blade for the specific application. Various efforts to optimize the blade configuration have been used over the years, with varying results.
By way of example, Zond Energy Systems, Inc. (the Assignee of the subject invention) has generally used a thinner airfoil configuration than its European counterparts. For example, the 34 meter blades manufactured by LM Glasfibre for Tacke""s 70 meter TW1.5s system employ a 39% thick section at the 25% spanwise location as compared t a 24% thick section on the comparable Zond Z46/48/50 blades. Even at 40% span, the airfoil section is 30% thick. This has a significant impact upon drag, reducing the energy capture from these blades by as much as 10%.
Turbines currently on the market have rotor loading of approximately 0.42-0.45 kW/m2 for machines certified IEC Class 1, 0.38-0.41 kW/m2 for Class 2 and 0.33-0.38 kW/m2 for Class 3. Taking a given wind turbine and then scaling both the rotor and drive train, including the generator, in proportion to each other it is a fairly straightforward series of calculations to determine the dependence of blade loads upon rotor size. If it is assumed that the rotor aerodynamics and solidity remain constant as the rotor is scaled, then the rated wind speed will remain constant for the various sized machines. For a turbine such as Zond""s 750 kW series, the tip speed of 85 m/s is approximately the higher limit. Using this as a fixed tip speed, in can be determined that the rated shaft speed will scale inversely to the rotor:
xcexa9rated=Vtip/Rxe2x80x83xe2x80x83(Equation 1),
where xcexa9rated is the rated shaft speed, Vtip is the fixed tip speed and R is the rotor radius.
Since the rated power, Prated, scales as the rotor diameter squared for a fixed rotor loading, and since the rated power is a product of the rated torque and the rated shaft speed, it follows that the rated torque scales as the cube of the rotor diameter:
xcexa9rated=Prated/xcexa9rated=xc2xd(xcfx81V3ratedCpxcfx80R2)(R/Vtip)xe2x88x92R3xe2x80x83xe2x80x83(Equation 2)
The rated torque results from the in-plane aerodynamic forces acting over the length of the blade. Mathematically, it results from the summation of the product of theses forces and the moment arm over the length of the blade:                               Q          rated                =                                            ∫                              r                hub                            R                        ⁢                                                            F                  x                                ⁡                                  (                  r                  )                                            ⁢              r              ⁢                              xe2x80x83                            ⁢                              ⅆ                r                                              =                                    R              2                        ⁢                                          ∫                                                      r                    hub                                    /                  R                                1                            ⁢                                                                    F                    x                                    ⁡                                      (                                          r                      /                      R                                        )                                                  ⁢                                  (                                      r                    /                    R                                    )                                ⁢                                  xe2x80x83                                ⁢                                  ⅆ                                      (                                          r                      /                      R                                        )                                                                                                          (                  Equation          ⁢                      xe2x80x83                    ⁢          3                )            
Where Fx represents the in-plane forces per unit length.
The mathematical model for all wind turbine airfoils follows these equations. What remains is to develop a better understanding of these models in order to maximize airfoil design. At present certain aspects of the design are not clearly understood and the resulting airfoil designs of the prior art are less than optimum.
The combination of aerodynamic optimization and structural optimization in accordance with the teachings of the invention results in a new and novel airfoil design having substantially improved performance characteristics of airfoil designs of the prior art. The aforementioned mathematic modeling yields maximum aerodynamic criteria. This is then coupled with a structural analysis to modify the optimum aerodynamic design into a balance, substantially optimized airfoil configuration. The resulting airfoils of the subject invention have substantial performance impact on GAEP when compared to the airfoils of the prior art. The subject invention is an airfoil design based on the theoretical optimum aerodynamic structure modified as required to maximize structural integrity.
The subject invention is the result of an effort to maximize and optimize airfoil configuration and design by determining the important characteristics of the mathematical definition of the airfoil consistent with the above prior-art recognized mathematical modeling.
This procedure provides the criteria for maximizing airfoil performance to achieve highest GAEP while taking into consideration the aerodynamic design parameters as balanced against structural requirements. The methodology of the subject invention permits the design of airfoils of predictable performance while achieving necessary structural integrity.
As a result of this approach, the subject invention has resulted in a family of airfoils having operational and structural characteristics with substantially enhanced performance capability over prior airfoils used in the same or similar applications. The family of airfoils includes thickness-to-chord ratios ranging from 14% to 45%.
In accordance with the invention, if the rotors are scaled up proportionately (i.e., the solidity remains constant), then substitution of Equation 2 into Equation 3 results in the conclusion that F, at any equivalent spanwise location (i.e., r/R) scales as the rotor diameter:
Fx≈Rxe2x80x83xe2x80x83(Equation 4)
For high lift-to-drag ratios, the in-plane forces in the outboard regions that dominate the structural loads result largely from the product of the dynamic pressure, the chord length, the lift coefficient, and the in-flow angle:
Fx=qratedcC1 sin "PHgr"xe2x80x83xe2x80x83(Equation 5)
where "PHgr" is the inflow angle.
Since the rotor is being scaled up, the chord, c, also scales as the rotor diameter. Since the rotor loading remains constant, qrated and sinxcfx86 remain constant, it follows from Equations 4 and 5 that C1 remains constant along the blade as they are scaled up. Since none of the flow angles or blade geometry changes other than being scaled up, it follows that the out-of-plane forces per unit length also scale as the rotor diameter:
Fx=qratedcC1 cos "PHgr"≈Rxe2x80x83xe2x80x83(Equation 6)
Therefore, the flapwise blade root bending moment Myrated also scales as the rotor diameter cubed:                               M          yrated                =                                            ∫                              r                hub                            R                        ⁢                                                            F                  y                                ⁡                                  (                  r                  )                                            ⁢              r              ⁢                              xe2x80x83                            ⁢                              ⅆ                r                                              =                                                    R                2                            ⁢                                                ∫                                                            r                      hub                                        /                    R                                    1                                ⁢                                                                            F                      y                                        ⁡                                          (                                              r                        /                        R                                            )                                                        ⁢                                      (                                          r                      /                      R                                        )                                    ⁢                                      xe2x80x83                                    ⁢                                      ⅆ                                          (                                              r                        /                        R                                            )                                                                                            ≈                          R              3                                                          (                  Equation          ⁢                      xe2x80x83                    ⁢          7                )            
In the subject invention, it has been determined that when the rotor is scaled up in diameter (keeping the solidity constant), while the rated power and tip speed both remain constant, the rated wind speed drops as the rotor diameter increases, according to the well-known relationship:
Prated=(xc2xd)xcfx81V3ratedCpxcfx80R2xe2x80x83xe2x80x83(Equation 8)
Assuming the rated power is constant, this yields:
Vrated≈Rxe2x88x92xc2xdxe2x80x83xe2x80x83(Equation 9)
This leads to the conclusion that the rated tip speed ration, X, increases with the rotor diameter:
X=(Vtip/Vrated)≈Rxe2x88x92⅔xe2x80x83xe2x80x83(Equation 10)
For a constant tip speed.
In this instance, Equation 2 becomes:
Qrated=Prated/xcexa9rated=Prated(R/Vtip)≈Rxe2x80x83xe2x80x83(Equation 11)
Substituting Equation 3 into Equation 11, this yields:
Fx≈1Rxe2x80x83xe2x80x83(Equation 12)
Which is dramatically different than previously assumed. Looking again at Equation 5, in the methodology of the subject invention the dynamic pressure at outboard station is dominated by the tangential velocity, so the drop in rated wind speed has little effect on the dynamic pressure at rated wind speed. Thus, the inflow angle varies inversely with the local speed ration and Equations 5 and 12 become:
Fx≈qratedcC1 sin "PHgr"≈RC1(1/X)≈RC1(1/X)≈C1R⅓xe2x80x83xe2x80x83(Equation 13)
Substituting Equation 10 for X, combining Equations 12 and 13 yields:
C1≈Rxe2x88x92{fraction (4/3)}xe2x80x83xe2x80x83(Equation 14)
at rated wind speeds. Substituting Equation 14 into Equation 6 yields
Fy≈qratedcC1 cos "PHgr"≈Rxe2x88x92⅓xe2x80x83xe2x80x83(Equation 15)
Fx and Fy represent the values at a given equivalent spanwise location, i.e., the same r/R location on each blade. Now, substituting Equation 15 into Equation 7, the result is:                               M                      y            ⁢                          xe2x80x83                        ⁢            rated                          =                                            ∫                              r                hub                            R                        ⁢                                                            F                  y                                ⁡                                  (                  r                  )                                            ⁢              r              ⁢                              xe2x80x83                            ⁢                              ⅆ                r                                              =                                                    R                2                            ⁢                                                ∫                                                            r                      hub                                        /                    R                                    1                                ⁢                                                                            F                      y                                        ⁡                                          (                                              r                        /                        R                                            )                                                        ⁢                                      (                                          r                      /                      R                                        )                                    ⁢                                      xe2x80x83                                    ⁢                                      ⅆ                                          (                                              r                        /                        R                                            )                                                                                            ≈                          R                              5                /                3                                                                        (                  Equation          ⁢                      xe2x80x83                    ⁢          16                )            
In the stated case, the root bending moment scales as the radius to the power 1.66.
This model was confirmed using Bladed and three rotors for a 750 kW turbine. (Bladed is a commercially available design program offered by Garrad Hassan). With a 50 m rotor, a 52 m rotor and a 55 m rotor. The measured scaling factor was in the range of 1.6, or very close to the calculated theoretical scale factor of 1.66 as derived in Equation 15). Realizing that the calculated analysis (Equation 16) ignores a number of secondary effects, e.g., the influence of the changing rated wind speed on the dynamic pressure and other variables, the actual test substantially verifies the calculated analysis. Recognizing that fatigue loads derive from the same aerodynamic model used in the above analysis, which calculates static (extreme) loads, it can be presumed that the fatigue loads will scale similarly as in Equation 16.
Using the assumption derived from the trade-off studies that halving the blade stiffness (doubling the deflection) results in a 15% reduction in fatigue, then Equation 16 can be used to calculate what size rotor will result in a 15% increase in loads. For example, using a Tackes""s TW1.5s turbine with a 70.5 m rotor and a 1,500 kW rating it can be determined from the trade-off studies that when the stiffness of the blade is halved that the loads will be reduced 15%.
The rotor can then be scaled up in accordance with the following formula, which is derived from Equation 16:                                                                                           (                                      M                    rated                                    )                                                  larger                  ,                  flexible                                            =                                                                    (                                          M                                              rated                        ⁢                                                  xe2x80x83                                                                                      )                                                        baseline                    ,                    flexible                                                  ⁢                                                      (                                                                  R                        larger                                            /                                              R                        baseline                                                              )                                                        5                    /                    3                                                                                                                          =                              .85                ⁢                                                      (                                          M                                              rated                        ⁢                                                  xe2x80x83                                                                                      )                                    baseline                                ⁢                                                      (                                                                  R                        larger                                            /                                              R                        baseline                                                              )                                                        5                    /                    3                                                                                                                          =                                                (                                      M                                          rated                      ⁢                                              xe2x80x83                                                                              )                                baseline                                                                        (                  Equation          ⁢                      xe2x80x83                    ⁢          17                )            
and, solving for the new rotor diameter yields:
Rlarger/Rbaseline=(1.85)/6=1.10xe2x80x83xe2x80x83(Equation 18)
Thus, the 70.5 m rotor could be increased by 10%, to a 77 m rotor without increasing the key design fatigue loads if the stiffness is relaxed two-fold. This results in a 20% increase in swept area and, assuming 50% of the energy capture comes from operation below rated, a 10% increase in net annual energy production. This example is for a rotor loading of 0.32 kW/m2 (for IEC Class 2).
As a result of this analysis, a number of initial rotor optimization studies were undertaken using a 77 m root diameter. This resulted in a final sizing analysis wherein a 1.8 MW machine, which is more cost effective, results in a scaling of the rotor diameter to 85 m. In the preferred embodiment, a more conservative 80.5 m rotor diameter is used.
As a result of these studies it has been determined that:
1. Airfoils up to 30% thick can be used at the first station (25%-30% radius) without significant loss in GAEP (Gross Annual Energy Production).
2. To maximize GAEP, the airfoil t/c (thickness) should not exceed 21%, 18% and 14%, for the second (55%-60% radius), third (75%-80% radius) and fourth blade stations (90%-95%), respectively.
3. Increasing the design cl increased the GAEP. Over the range of design c1 a value of 1.25 was found to be the optimum for all blade stations. A lower design c1 in the tip region is beneficial for keeping the blades out of stall.
4. The loss in GAEP from increasing the airfoil t/c along the blade can be easily compensated by increasing the design c1. Thicker airfoils can be used without sacrificing energy capture.
These aerodynamic studies have been combined with structural design studies to provide a comprehensive design criteria wherein the thickness and lift range of the airfoils is optimized. For optimum aerodynamic performance, thin airfoils having high lift-to-drag ratios are desired while thick airfoils are favorable for structural reasons. High-lift airfoils yield larger lift-to-drag ratios for a given amount of laminar flow as compared with low-lift airfoils, which increases energy capture. High-lift airfoils also have, however, structural implications. Consequently, a balance between aerodynamic and structural considerations is required for defining the optimum airfoil t/c and lift range for a particular blade.
Thus, it is desirable to quantify the effects of airfoil thickness (t/c) and lift range on energy capture. A tradeoff study using the 1.63-MW NGT having a 77 m rotor was used to quantify this information. The analysis yielded data quantifying the effect of airfoil t/c and lift range on the GAEP for a single blade segment at four different radial positions. Entire blades were then designed for maximum energy capture, using the most promising airfoil t/c and lift range for each of the four selected stations. The effects of truncating the inboard chords of the blades on energy production provided data establishing the impact of minimizing the blade area in the root region. For a given set of airfoils it was found that a reduction in chord yields a decrease in physical thickness that is not desired structurally unless the airfoils are also truncated.
In the preferred embodiment the blade geometry is designed for maximum annual energy production. In designing each blade segment, the optimum axial inflow of 113 and design lift coefficient c1 are prescribed and the corresponding chord and twist/pitch are obtained using the inverse design capability of the computer program PROPID (PROPID is a commercially available design program. A selected design c1, which is the c1 for which maximum lift-to-drag ratio is achieved, results in the same chord length independently of the airfoil t/c considered. In the example used to confirm this analysis the following design constraints were used:
Mechanical rated power of 1.8 MW.
System efficiency of 90%, yielding an electrical rated power of 1.62MW
Three-bladed, upwind rotor having a diameter of 77 m
Design tip-speed ratio of 7.68, which corresponds to a tip-speed of 80 m/s at rated power
Sea level atmospheric conditions.
In order to determine the annual energy production on this model, an IEC wind class II (average wind speed of 8.5 m/s at hub height) and a Rayleigh wind speed distribution were considered. No losses were taken into account apart from the 90% system efficiency. The GAEP was computed at 100% availability.
The design process was carried out on blade segments at four radial stations, namely 25%-30% radius, 55%-60% radius, 75%-80% radius, and 90%-95% radius. The design c1 is prescribed to 1.05 and the airfoil thickness is varied for each of the four radial stations. Also, the airfoil thickness at each radial station was fixed and the design c1 was varied. For this study on the effects of the design c1 on energy capture, the airfoil t/c was fixed at 27% for station 1, 21% for station 2 16% for station 3 and 12% for station 4. Entire blade design is also considered with the airfoils used along the blade defined at the same four stations as the segment designs and the hub modeled as a cylinder.
Out of this study, a baseline case has been developed representing a best-case scenario in terms of maximizing energy capture as it uses higher-lift airfoils than those of the prior art Z-48 blade, while having similar airfoil t/c distribution.
Out of this it has been determined that airfoils up to 30% thick can be used at the first station (25%-30% radius) without a significant loss in GAEP. Truncating the 30% thick airfoil significantly reduces GAEP with losses up to 12 times greater than that for the non-truncated 30% airfoil. These losses in GAEP from truncating the 30% thick airfoil may be weighted against the structural advantages that truncation provides.
GAEP is maximized when the airfoil t/c does not exceed 21% 18% and 14%, for the second 55%-60% radius), third (75%-80% radius), and fourth blade stations (90%-95%), respectively.
Increasing the design c1 increases the GAEP. A value of 1.25 has be found to be the optimum or all blade stations although a lower design c1 in the tip region might be required depending on the ability of the controller to keep the blades out of stall.
The loss in GAEP from increasing the airfoil t/c along the blade can be easily compensated by increasing the design c1. Therefore, thicker airfoils than those of the prior art Z-48 blade can be used without sacrificing energy capture.
Truncating or shortening the inboard chord should be limited to 25%-30% of the maximum nominal chord length value. Such truncation of the chord has only a small effect on the GAEP, particularly if the root airfoil is not truncated.
Based on this criteria, airfoil design may be optimized using a balance of maximized aerodynamic and maximized structural requirements to provide a dependable, efficient airfoil with enhanced GAEP over prior art configurations.
It is, therefore, an object and feature of the subject to provide a means and method for designing an enhanced airfoil configuration for a wind turbine maximizing aerodynamic design parameters.
It is another object and feature of the subject invention to provide a means and method for designing an enhanced airfoil configuration for a wind turbine maximizing structural design parameters.
It is a further object and feature of the subject invention to provide a means and method for designing an enhanced airfoil configuration with balanced aerodynamic and structural characteristics.
It is an additional object and feature of the subject invention to provide an airfoil designed with enhanced GAEP capability.
Other objects and features of the invention will be readily apparent from the accompanying drawings and detailed description of the preferred embodiment.